We formulate a finite element method for the computation of solutions to an anisotropic phase-field model for a binary alloy. Convergence is proved in the H1-norm. The convergence result holds for anisotropy below a certain threshold value. We present some numerical experiments verifying the theoretical results. For anisotropy below the threshold value we observe optimal order convergence, whereas in the case where the anisotropy is strong the numerical solution to the phase-field equation does not converge.